Theorem 3.28 [1]}

Let \(\mathbf {U}_\mathbf {n} = (U^{(1)}_{\mathbf {n}},\dots ,U^{(\ell )}_{\mathbf {n}})\)  be \(\ell \)  generalized U-statistics, on common samples \(\lbrace X_{{\text{\bfseries \ttfamily x}}1},\dots ,X_{{\text{\bfseries \ttfamily x}}n_{\text{\bfseries \ttfamily x}}}\rbrace \)  where \(n_{{\text{\bfseries \ttfamily x}}}/n \rightarrow p_{{\text{\bfseries \ttfamily x}}} > 0\)  for every \({\text{\bfseries \ttfamily x}}\)  as \(n = |\mathbf {n}| \rightarrow \infty \) . Then the vector
\( \mathbf {Z}_{\mathbf {n}} \;:=\; \sqrt{n} \left(\mathbf {U}_\mathbf {n} - \operatorname{\mathbb {E}}\mathbf {U}_\mathbf {n} \right) \;\;\xrightarrow[\;\text{\rm in distribution }\;]{n \rightarrow \infty }\;\; \mathbf {Z}\) 
